Coloring by Pushing Vertices
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909604626038784 |
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| author | Rautenbach, Dieter Schwartze, Laurin Werner, Florian |
| author_facet | Rautenbach, Dieter Schwartze, Laurin Werner, Florian |
| contents | Let $G$ be a graph of order $n$, maximum degree at most $Δ$, and no component of order $2$. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of $G$ as a function $ρ:V(G)\to\mathbb{N}_0$ for which $$σ:V(G)\to\mathbb{N}_0:u\mapsto \left(1+ρ(u)\right)d_G(u)+\sum_{v\in N_G(u)}ρ(v)$$ is a vertex coloring, that is, adjacent vertices receive different values under $σ$. They show the existence of a proper pushing scheme $ρ$ with $\max\{ ρ(u):u\in V(G)\}\leq Δ^2$ and conjecture that this upper bound can be improved to $Δ$. We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme $ρ$ with $\sum_{u\in V(G)}ρ(u)\leq \left(2Δ^2+Δ\right)n/6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05252 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coloring by Pushing Vertices Rautenbach, Dieter Schwartze, Laurin Werner, Florian Combinatorics Let $G$ be a graph of order $n$, maximum degree at most $Δ$, and no component of order $2$. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of $G$ as a function $ρ:V(G)\to\mathbb{N}_0$ for which $$σ:V(G)\to\mathbb{N}_0:u\mapsto \left(1+ρ(u)\right)d_G(u)+\sum_{v\in N_G(u)}ρ(v)$$ is a vertex coloring, that is, adjacent vertices receive different values under $σ$. They show the existence of a proper pushing scheme $ρ$ with $\max\{ ρ(u):u\in V(G)\}\leq Δ^2$ and conjecture that this upper bound can be improved to $Δ$. We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme $ρ$ with $\sum_{u\in V(G)}ρ(u)\leq \left(2Δ^2+Δ\right)n/6$. |
| title | Coloring by Pushing Vertices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.05252 |